Let P(x) be a polynomial of degree n with leading coefficient 1 . Let v(x) be any function and v1(x)=∫v(x)dxv2(x)=∫v1(x)dx…vn+1(x)=∫vn(x)dx then ∫P(x)v(x)dx is equal to

Integrating by part repeatedly , we get ∫P(x)v(x)dx=P(x)v1(x)−∫P′(x)v1(x)dx=P(x)v1(x)−P′(x)v2(x)+∫P′′(x)v2(x)dx=P(x)v1(x)−P′(x)v2(x)+P′′(x)v3(x)−….+(−1)n∫P(n)(x)vn(x)dxNow , use P(n)(x)=n!